Let \(\mathrm{f(x) = (x + \sqrt{6})^2 - (x - \sqrt{6})^2}\). Which of the following is equivalent to \(\mathrm{f(x)}\)?

1. INFER the most efficient approach

• Given: \(\mathrm{f(x) = (x + \sqrt{6})^2 - (x - \sqrt{6})^2}\)
• Key insight: This is a difference of squares pattern \(\mathrm{a^2 - b^2}\) where:
  • \(\mathrm{a = x + \sqrt{6}}\)
  • \(\mathrm{b = x - \sqrt{6}}\)
• Strategy: Use the difference of squares identity rather than expanding each term individually

2. SIMPLIFY using the difference of squares formula

• Apply \(\mathrm{a^2 - b^2 = (a + b)(a - b)}\)
• Calculate the factors:
  • \(\mathrm{a + b = (x + \sqrt{6}) + (x - \sqrt{6}) = x + \sqrt{6} + x - \sqrt{6} = 2x}\)
  • \(\mathrm{a - b = (x + \sqrt{6}) - (x - \sqrt{6}) = x + \sqrt{6} - x + \sqrt{6} = 2\sqrt{6}}\)

3. SIMPLIFY the final multiplication

• \(\mathrm{f(x) = (a + b)(a - b) = (2x)(2\sqrt{6}) = 4x\sqrt{6}}\)
• This matches choice (C)

Answer: C. \(\mathrm{4x\sqrt{6}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the difference of squares pattern and instead choose to expand both squares individually, leading to more complex calculations and increased chance of algebraic errors.

When expanding \(\mathrm{(x + \sqrt{6})^2}\) and \(\mathrm{(x - \sqrt{6})^2}\) separately, students often make sign errors, particularly when distributing the negative sign across \(\mathrm{(x - \sqrt{6})^2}\). They might incorrectly write:

\(\mathrm{f(x) = (x^2 + 2x\sqrt{6} + 6) - (x^2 + 2x\sqrt{6} + 6) = 0}\)

This may lead them to select Choice A (0).

Second Most Common Error:

Poor SIMPLIFY execution: Students recognize the difference of squares but make errors in calculating \(\mathrm{a + b}\) or \(\mathrm{a - b}\). A common mistake is:

• \(\mathrm{a - b = (x + \sqrt{6}) - (x - \sqrt{6}) = \sqrt{6} - \sqrt{6} = 0}\) (forgetting to distribute the negative)

This leads to \(\mathrm{f(x) = (2x)(0) = 0}\), causing them to select Choice A (0).

The Bottom Line:

This problem tests whether students can recognize algebraic patterns (difference of squares) and execute multi-step algebraic manipulations accurately. The key insight is seeing the structure rather than diving into lengthy expansions.